JEE Main 2024 · 30 Jan 2024 Shift 1

If the domain of the function f(x) = cos⁻¹((2 − |x|)/4) + {logₑ (3 − x)}⁻¹ is [−α, β) − {γ}, then α + β + γ is equal to :

1. A8
2. B9
3. C11
4. D12

If z = x + iy, xy ≠ 0, satisfies the equation z² + i z̄ = 0, then |z²| is equal to :

1. A1/4
2. B1
3. C4
4. D9

Consider the system of linear equations x + y + z = 4μ, x + 2y + 2λz = 10μ, x + 3y + 4λ²z = μ² + 15, where λ, μ ∈ R. Which one of the following statements is NOT correct ?

1. AThe system is consistent if λ ≠ 1/2
2. BThe system is inconsistent if λ = 1/2 and μ ≠ 1
3. CThe system has unique solution if λ ≠ 1/2 and μ ≠ 1, 15
4. DThe system has infinite number of solutions if λ = 1/2 and μ = 15

If f(x) = | 2cos⁴x 2sin⁴x 3 + sin²2x; 3 + 2cos⁴x 2sin⁴x sin²2x; 2cos⁴x 3 + 2sin⁴x sin²2x |, then 1/5 f′(0) = is equal to :

1. A0
2. B1
3. C2
4. D6

Two integers x and y are chosen with replacement from the set {0, 1, 2, 3, ..., 10}. Then the probability that |x − y| > 5, is :

1. A60/121
2. B30/121
3. C62/121
4. D31/121

Let Sₙ denote the sum of first n terms of an arithmetic progression. If S₂₀ = 790 and S₁₀ = 145, then S₁₅ − S₅ is :

1. A405
2. B395
3. C390
4. D410

The maximum area of a triangle whose one vertex is at (0, 0) and the other two vertices lie on the curve y = −2x² + 54 at points (x, y) and (−x, y), where y > 0, is :

1. A92
2. B108
3. C88
4. D122

Let f : [−π/2, π/2] → R be a differentiable function such that f(0) = 1/2. If the limₓ→₀ [x ∫₀ˣ f(t)dt]/(eˣ² − 1) = α, then 8α² is equal to :

1. A1
2. B2
3. C4
4. D16